Walmart's carries some products in their boating section that may help, I've seen some 3M polishes there. Any mild polishing compound should remove the oxidation, if not by hand then with an electric buffer. The buffer should be used with caution, as it can remove a lot very quickly depending on how abrasive the compound is and how much pressure is applied to the surface. I don't think a one time polishing with a buffer would jeopordize the gelcoat, but try hand polishing first.
After you bring up the shine to your satisfaction, apply a coat of carnuba based wax to the finish, then go show off your boat!
Brian
You're never too old to be immature.